We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$ and denote by $x_1 \leq \ldots \leq x_{n+m}$ the roots of $P$ counted with multiplicity, with $Q = \prod_{k=m+1}^{n+m}(X-x_k)$ and $R = \prod_{k=1}^{m}(X-x_k)$.
Verify that for all $x \in ]-\infty, -1[$, we have $|Q(x)| > |Q(-1)|$.